A283298 Diagonal of the Euler-Seidel matrix for the Catalan numbers.
1, 3, 26, 305, 4120, 60398, 934064, 15000903, 247766620, 4182015080, 71816825856, 1250772245698, 22039796891026, 392213323252200, 7038863826811100, 127248841020380105, 2315130641074743540, 42358284517663463380, 778876539384226875800
Offset: 0
Links
- Paul Barry and A. Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010) # 10.8.2, Example 11.
Programs
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Maple
A000108 := n-> binomial(2*n, n)/(n+1): A283298 := proc(n) add(binomial(n,i)*A000108(n+i),i=0..n) ; end proc: seq(A283298(n),n=0..30) ;
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Mathematica
Table[Sum[Binomial[n, i] CatalanNumber[n + i], {i, 0, n}], {n, 0, 50}] (* Indranil Ghosh, Jul 20 2017 *) -
PARI
C(n) = binomial(2*n,n)/(n+1); \\ A000108 a(n) = sum(i=0, n, binomial(n,i) * C(n+i)); \\ Michel Marcus, Nov 12 2022
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Python
from sympy import binomial, catalan def a(n): return sum(binomial(n, i)*catalan(n + i) for i in range(n + 1)) print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 20 2017
Formula
a(n) = Sum_{i=0..n} binomial(n,i) * A000108(n+i).
D-finite with recurrence 2*n*(2*n+1)*(9*n-11)*a(n) +(-711*n^3+1589*n^2-986*n+144)*a(n-1) -10*(n-1)*(9*n-2)*(2*n-3)*a(n-2)=0.
a(n) ~ 2^(2*n) * 5^(n + 3/2) / (27 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 01 2025